The monodromy of meromorphic projective structures

Abstract

We study projective structures on a surface having poles of prescribed orders. We obtain a monodromy map from a complex manifold parameterising such structures to the stack of framed PGL 2 ⁡ ( C ) \operatorname {PGL}_2(\mathbb {C}) local systems on the associated marked bordered surface. We prove that the image of this map is contained in the union of the domains of the cluster charts. We discuss a number of open questions concerning this monodromy map.